It reflects the power Christ gave to pray in His name... ...the Lord Jesus Christ instituted prayer in His Name. He promised that such prayer will be particularly effectual.
St. Ignatius Brianchaninov On the Prayer of Jesus

The Name of Jesus is the sole source of salvation...
Nor is there salvation in any other, for there is no other name under heaven given among men by which we must be saved.
St. Paul in Acts 4:12

The Jesus Prayer will overcome negative spiritual emotions...
When during prayer your heart is overwhelmed with despondency and melancholy, be sure that these proceed from the Devil, endeavoring by every means to hinder you in your prayer. Be firm, take courage, and by the remembrance of God drive away this deadly feeling.
St John of Kronstadt (d. 1908) On Prayer

It is the only sure means of a close relationship with Christ...
Those who have truly resolved to serve God must practice the remembrance of God and unceasing prayer to the Lord Jesus Christ, saying with the mind: Lord Jesus Christ, Son of God, have mercy on me, a sinner.” By this practice, while guarding oneself from distraction and while maintaining peace of conscience, one can draw near to God and be united with Him. Other than by unceasing prayer, according to the words of St Isaac the Syrian, it is impossible to draw near to God.
St Seraphim of Sarov (d.1833) quoted in On the Prayer of Jesus

It is the surest means to gain Heaven...
...the saints who
for their entire life kept in their heart the Jesus Prayer and the presence of
our Lord Jesus Christ. Only they ascend directly to Paradise, for their
souls became a paradise while on earth. Where the Lord God is—there is
Paradise. Know that if a person trains himself to say the Jesus Prayer continually,
when his hour of death comes, Christ Himself will take his soul
and it will abide with Him forever.From the vision of the Nun Tatiana of St. Petersburg in the early 20th century

Sanctification comes with the repeated praying of the prayer...
The name of our Lord Jesus Christ, descending into the depths of the heart, will subdue the serpent holding sway over the pastures of the heart, and will save our soul and bring it to life. Thus, abide constantly with the name of our Lord Jesus Christ, so that the heart swallows the Lord and the Lord the heart, and the two become one. But this work is not done in one or two days; it needs many years and a long time. For great and prolonged labor is needed to cast out the foe so that Christ dwells in us.
St John Chrysostom (d. 407) in Writings from the Philokalia